The Kegel–Wielandt $\sigma$-problem: reduction to simple groups

We reduce the solution of the Kegel–Wielandt $\sigma$-problem for an arbitrary partition $\sigma$ of the set of all primes to its solution in the class of all $\sigma$-complete simple nonabelian groups. We also give a sufficient criterion for the $\sigma$-subnormality of a subgroup in a finite group for a partition $\sigma$ in which 2 and 3 belong to the same component.